Electrical generator of products and functions



April 6, 1954 E. LAKATOS 2,674,409 ELECTRICAL GENERATOR 0F PRODUCTS ANDFUNCTIONS Filed July 12 1950 6 Sheets-Sheet l INVENTOR E. LA/(A 7'05 8VAT TORNE V E. LAKATOS ELECTRICAL GENERATOR OF PRODUCTS AND FUNCTIONSFiled July 12. 1950 6 Sheets-Sheet 2 INVENTOR E. LAKATOS By m ATTORNEYApril 6, 1954 E. LAKATOS 2,674,409

ELECTRICAL GENERATOR 0F PRODUCTS AND FUNCTIONS Filed July 12, 1950 6Sheets-Sheet 3 s u T -s a a k k g x a G FIG. 4

A 7' TORNEV E. LA KATOS ELECTRICAL GENERATOR OF PRODUCTS AND FUNCTIONSFiled July 12, 1950 6 Sheets-Sheet 4 M; Ki

lNl/ENTO/ By E. LA/(A T05 MM AT TORNEV E. LAKATOS pril 6, 1954ELECTRICAL GENERATOR OF PRODUCTS AND FUNCTIONS Filed July 12, 1950 6Sheets-Sheet 5 INVENTOR E. LAKATOS BY E A 7' TORNE Y April 6, 1954 E.LAKATOS 2,574,409

I ELECTRICAL GENERATOR OF PRODUCTS AND FUNCTIONS Filed July 12. 1950 6Sheets-Sheet 6 /N VENT OR E. LA/(ATOS BY ATTORNEY Patented Apr. 6, 1954ELECTRICAL GENERATOR OF PRODUCTS AND FUNCTIONS Emory Lakatos, Cranford,N. .L, assignor to Bell Telephone Laboratories, Incorporated, New York,N. Y., a corporation of New York Application July 12, 1950, Serial No.173,271

16 Claims. 1

This invention relates to electrical computing networks producingquantities of electrical energy representing functions of a singlemathematical quantity, or of a plurality of mathematical quantitles.

The object of the invention is an electrical network producing variousquantities of electrical energy which, when algebraically added, willproduce a quantity of electrical energy representing a function of asingle mathematical quantity, or a function of a plurality ofmathematical quantities.

A feature of the invention is the use of a plurality of triangular wavesproducing electrical currents which are respectively biased by otherelectrical currents representing functions of one or more mathematicalquantities, then rectified, and the products of the variousrectifications algebraically added to produce a quantity of electricalenergy representing a function of one, or more, of the mathematicalquantities, Specifically, the sum of the products of the various.rectifications may represent the square, or square voltage, and theresultant wave is rectified, the

rectified output includes a current which is a linear function of thesquare of the bias voltage. If the rectifier is associated with areversefeedback amplifier, the output of the amplifier will be a function ofthe square root of the bias voltage. Also, if there are two triangularwaves, one biased with the sum, and the other with the difference, oftwo bias voltages, and the resultant waves are rectified, the rectifiedoutputs include currents which are linear functions of the product ofthe bias voltages. If these rectifiers are associated with a reversefeedback amplifier, the output of the amplifier will be a function ofthe quotient of the bias voltages.

The invention will be more easily understood r from the followingdescription of some typical embodiments of the invention; and from thedrawings, in which:

Figs. 1 and 1A schematically show simple networks embodying theinvention;

in Fig. 2.

of e the voltage applied to the rectifier.

Fig. 2 graphically shows the wave shape of the generator in Fig. 1;

Figs. 3, 3A, 4, 4A graphically show the voltage and current wave shapeswith a diode rectifier;

Fig. 5 schematically shows a network producing a quantity of electricalenergy representing the product or quotient of two factors;

Fig. 6 schematically shows a simplified network producing a quantity ofelectrical energy representing the product of two factors; and

Fig. '7 schematically shows a network produc ing a quantity ofelectrical energy representing the square, or square root, of amathematical quantity.

Fig. 1 shows schematically a rectifier l0, having a plate I! and acathode l2, the latter being shown grounded for illustrative purposes.The rectifier I0 is driven by a generator I4 through an externalresistance Rg. The generator delivers a voltage e(t) whose wave shape isshown This wave has two principal components. One is a repeatedtriangular wave of period tn and half amplitude Em volts, and the otheris a bias voltage V. Although V is shown as a constant voltage, it maybe allowed to vary with time, provided that its variation is slowrelative to one period of the triangular wave. Assuming that thiscondition is satisfied, the wave 'e(t) will have a maximum amplitude inthe positive direction and Em+V in the negative direction. Weshall beinterested in calculating the current 1'.(t) that flows in response toe(t) "and more particularly, its average value 2.

To do this, in Fig. 3, consider the dashed curve labeled i:F(e Thisshows the forward current through an electronic rectifier as a functionFor positive values of e the current voltage relationship in somerectifiers may be slightly curved, and in other rectifiers thisrelationship may be substantially linear, but, for simplicity ofdiscussion, the relationship may be idealized as substantially linear,the slope m1 of the curve being the reciprocal of the average resistanceof the rectifier, RD, in the forward direction. This average resistanceusually is low, not exceeding a few hundred ohms. This curve may ariseat the origin, but, in the case of electronic diodes, the origin of thecurve is usually not at the origin. For small voltages, which may benegative, the relationship between current and voltage is curvilinear,and eventually, for some voltage, the

current is substantially zero. The effective cutofi point C may bedefined as the point where the idealized straight line portion of thecharacteristic, when extended, cuts the voltage axis.

If a resistance Hg is inserted in series with the diode and if R; isvery large compared to the forward resistance of the diode, the over-allcurrentvoltage relation i=G(E) will be approximated very closely by astraight line which starts at the effective cut-ofi point C and has aslope Since for practical applications Rg may be taken at least 100times Rn, the slope may be taken to be m=1/R We note for futurereferencethat if the diode has an intercept i amperes on the currentaxis, the over-all characteristic has, on the same axis, an intercept b:0 D 0 D R +R, R, and on the voltage axis an intercept T11R or io'RD.

Suppose now that the'voltage E applied is specifically that shown inFig. 2, namely e(t). A plot of this voltage against time is that labeleda'b'cd' in Fig. 3. The resulting current flow has the wave shape i(t),labeled abcd in Fig. 3A, provided that Wl l l There will be current flowfor a time interval tr during each cycle.The amplitude of the triangularwave, that is, the height of the triangle from the base to to b is 2Em.The height of the triangle abc' is (V+Em+bRg). The length of line a'c'is Thelength of this intervalis the same as that from a to c and has thevalue r ll( m+ mmS S m The maximum current is in. and its value is verynearly The mean current Zis obtained by averaging the current i(t) overone cycle. As the current i(t) has a triangular variation, the averagecurrent A second case that has to be considered is the one where theplate and cathode connections are reversed with respect to generator andground. This is shown in Fig. 1A by the rectifier [0, its plate I l andcathode l2. The instantaneous an average currents will be denoted by iand 1 respectively. Since conduction is from plate to cathode, it willbe seen for this case that the flow of z" is in the opposite sense tothe flow of 1'. Since the plate is maintained at ground potential, thecathode need not get more positive than, say, +1 volt to haveconduction. The plot of i against the cathode volta is therefore of theform shown in Fig. 4A. It will be seen that the complete cuirent-voltagerelation. i=G(E) is merely the mirror image of the one shown in Fig. 3.If the applied voltage, wbcd', has a maximum positive amplitude of Em+Vand a maximum negative amplitude -E1n+V (with -Em VEm), the resultingcurrent wave i'(t) is that shown by about in Fig. 4A. The duration ofthe wave is its maximum current amplitude is i'm and its value is verynearly i,,.=(E,,,VbR,)/R, (6)

and the average current is i='(E,,,-VbR,) /4E,,.R, (7)

In view of the fact already stated that 12 and 1" flow in oppositedirections, we may write i=i' and then we have The results contained inEquations 4 and 8 will be needed in the description of the functioningof the present invention.

Fig. 5 is a preferred embodiment of a multiplier. The two variables xand 11 whose product is desired are represented by proportionate signalvoltages 2D and 23, respectively. These voltages may be convenientlythought of as being slowly varying direct-current voltages, but they canalso be alternating voltages having a period long compared to the periodof the triangular wave. It is to be assumed that the internal impedanceof the generators furnishing these voltages is quite small, say a fewohms; and it is, of course, evident that the voltages representing a:and 11 may be obtained, if desired, by suitably fractionalizing thevoltage from a common source.

The a: voltage 2|] is applied to a shunt feedback amplifier 2| of thetype disclosed in Patent 2,401,779, issued June 11, 1946 to K. D.Sw'artzel. The output of this amplifier is a negative copy, -a: of itsinput. Similarly, the y voltage 23 is applied to amplifier 24 whoseoutput is then -y volts. The output impedances of these amplifiers arevery low, say, a few ohms.

These voltages 0:, -:c, y and y are inputs to the multiplier proper.Another input voltage that is required for the operation of the deviceis a periodic triangular wave which comes in over lead 22. In slope, itis symmetrical about the time axis and has a half amplitude of em. Itsperiod is to. Its rates of rise and fall may be approximately equal. Thegeneration of this triangular wave may be conveniently accomplishedthrough the use of a square wave generator of any conventional type. Theoutput of 85 is fed to the input of integrator 86, which may consist ofa high gain amplifier 81 having an input resistance 88 and a feedbackcapacitor 89, as shown, for example, in British Patents 5 75,250,February 11, 1946, Cossor Limited and 580,527, September 11, 1946, A. D.Blumlein. I The amplitude of this wave may be made any desirable value,say volts, by proper choice of the amplitude of the square wave and theRC product of the integrator. Making the duration of the positive andnegative portions of the square wave approximately equal satisfies therequirement that the rates of rise and fall of the triangular wave areto be approximately equal.

The five voltages, namely, the triangular waves, is: and :y, are appliedto the diode pairs 50A, 50B and 5| A, 5| B through four groups of threeresistors in the following manner. The-plate 5 52A of diode 50A isconnected at junction 10 to resistors 30, 3| and 40. The plate 523 ofdiode 50B is connected at junction H to resistors 32, 33 and 4|. Thecathode 55A of diode 5|A connects at junction 12 to resistances 34, 35and 42 while the cathode 55B of diode SIB connects at junction 13 toresistances 36, 31 and 43.

The other end of resistances 40, 4|, 42 and 43 are connected to lead 22coming from the triangular wave generator. The signal voltage as isapplied to the resistances 30 and 34. The voltage --a: is applied toresistances 32 and 36. The voltage 11 is applied to resistances 3| and31 and its negative, -y, is applied to resistances 33 and 35.

The resistances 40, 4|, 42 and 43 may be taken at any convenient valueRu, say 100,000 ohms. They should, however, be as nearly identical invalue as is feasible in order to obtain the highest accuracy from themulti lier.

Likewise, the resistances 30 through 31, inclusive, are to be as nearlyalike in value as feasible. While not absolutely necessary it isdesirable that their nominal value R1 be taken the same as for theresistances 40 through 43, inclusive. In what follows we shall assumethat this is done. I

The currents from the cathodes 53A, 53B and from the plates 54A, 54B arefed to the junction 61 which connects to the first grid 66 of ahighgain, shunt feedback amplifier 60, having a feedback resistance 6| ofvalue R2. This amplifier may be of the type disclosed in United StatesPatent 2,401,779, issued June 11, 1946, to K. D. Swartzel. It is a.property of this amplifier that its effective input impedance is verylow, of the order of a few ohms. Hence, in sofar as the flow of thecurrents i1, i2, is" and i4" is concerned, the junction 61 isessentially at ground potential.

Connected to the junction 61 is a capacitor 68. The function of thiscapacitor is to perform the time averaging operation on, the currentsflowing into or out of the junction 61. This implies that the capacitor68 effectively filters out those components of the current waves,'suchas ME) in Figs. 3 and 4 which have periods to orless. The frequencies inthe signals :2 and y are not attenuated to any appreciable extent.Consequently I, the average of the currents flowing "into the junction01 and thence into the grid 66 is the same as the sum of the timeaverages of the currents i1, i2, i3" and i4 actually present. That isT=t+e+e"+a" (9) Also connected to junction 6'! is a bias circuit Thisconsists of a center tapped potentiometer 62, energized by batteries 64and 65, poled as shown. The slider 63 of the potentiometer connects tojunction 6'! through a resistance 60 of any convenient value, say R0.The current flowing into junction 31 from the slider 63is 6b/R0, whereeb is the voltage at the slider. The purpose of this bias circuit is topermit the zeroing of the output 1) of amplifier 60, as explainedhereinbelow.

The output voltage v isrelated to the time average of all the currentsentering the junction 61 by the relation 2( b/ v 2( 1+ 2+ 3"+ 4"F b/ 0)Using Relations 4, 8 and 10, the value of the '6 whose positiveamplitude is (em+a:+y)/3. The circuit at junction 10 and the diode 50Aare fully equivalent to the circuit of Fig. 1 with resistance Rgconnected to diode ID. The mean current for this case is given byEquation 4. Hence we identify Em with 8112/3 and V with (23+?!) /3.

For junction H and diode 503, the conditions are the same except that Vis now to be identified with -(x+y)/3.

The circuit at junction 12 and diode 5|A are fully equivalent to thecircuit of Fig. 1 with the resistance Rg connected to diode H). The meancurrent for this case is given by Equation 8. We identify Rg with RIO/3,Em with em/3 and V with (a:y) /3. The conditions at junction 13 aresimilar except that V is to be identified as (.r+y) /3.

We also note that the intercept b in Figs. 3 and 4 may have differentvalues for each of the diodes. We therefore assume that for diode 50A itis b1, for 503 it is be, for EIA it is 133, and for 5|B it is b4. Then,using the above identifications, substituting the values into Equation 4or 8 as required, we get Substituting these values into (10) andcollecting terms, gives Then for non-zero values of signal voltages,Equation 15 reduces to v=- (sxy+2xRo b1b, 3+bo+ e Rg Z/ 0( 1 2+ 3 4))The error introduced by the still remaining terms can be evaluated asfollows. As already stated the current intercepts in through b4 arerelated to the typical current intercepts in of the diode characteristic(see 3) through the approximate relation b=eRD/R. The intercept 2'0varies from tube to tube due to manufacturing variations. Thisvariationmay;

befiaslarge as 2 10 per. cent. v="If We take an extremcacase such-thatb1=b4=1l0m average value, nnd" bz. .-=ba=0.9.0x average, .the:error=term= is 233130 (0.4 'iORD/Rg). The largest error will occurwhen::c=em/ 2, and for a typical diode in which, Rv=250 ohms and i is10*. amperes; since JBneBR the, extreme error is proportional to -l'.2'l0- 250em. The error relative to the corresponding maximum usefulsignal,- namely 8x11 rior 261x1 iswtherefore 0.l5/em. As a practicalwaluefor em is 100 volts, even in the most extreme case, the errorreferred to'the maximum is only .0215 per cent and on the average itwill be much less'thanthis. Furthermore by careful selection :an'dmatching'of the diodes, the error can be made'very nearlyzero. Then, forall practical purposes, the relation between the output voltage v'ofamplifier 60 and the-input signals at and 1/ maybe written as Asstated before, a practical value of 8m is 100 volts. Recalling thatneither :0 nor y may exceed rem/2, that is, 50 volts in thiscase, themaximum value of 1) will be 50 volts if we take the feedback resistanceR2 (resistance Bl) as equal to R0. For. other applications, any desiredscale factor may be obtained by proper choice of Rz/Ro.

While the multiplier has been disclosed as using electron tube diodes,it is evident that each of these may be replaced by any solid rectifier,preferably the high back. voltage germanium crystal diode. Since-thecurrent voltage characteristic of such crystal passes through theo'ri'girifthe intercepts bi, b2, in, bi, arezero, thus the biaspotentiometer 62 may be eliminated.

With 11:0, and V=(::c :11) Equations 4 and 8 may be respectively writtenas E QUAIION 4a 1 1/ Em 2Em-I 2Emy I 11 2m EQUATION 8a -..'I?he first,fourth, eleventh, and twelfth lines may respectively be identified withEquations 11, 12, 13, 14. Other typical networks, using differentcombinations of currents, will be described herieinbelow; It is: evidentthat many other net..- -works may be devised within the scope of theinvention to produce quantities of electrical energy representingvarious functions .of the quantities a: and y; involving in each casethe use of the auxiliary triangular function E(t'). The network to beused in any particular case will usually be determined by engineeringconsiderations, such as the relative simplicity of 'wiring and minimumuse of apparatus.

Fig. 6 shows a form of the multiplier'in' which the need for supplyingnegative copies of the a: and 1/ signal voltages has been eliminated.This is desirable at times for reasons of circuit economy. In thisfigure, everything to the right of junctions 10, 'H, 12 and "isidentical with the corresponding elements of Fig. 5. The square wavegenerator and the integrator 85, are also the same as before. Thesources of signal 'voltages 3:, 20, and y, 23, are also unchanged, asare the resistive elements 30, 3|, 34, 36, 40, 42 and 43. These will beassumed to be alike in value, of R0 ohms.

' The signal a: is applied to resistors 30 and 34. The signal 11 isapplied to resistors 3| and 36. The triangular wave is applied toresistors 40, 80, 42 and 43. Resistors 8i and 82, of valueRa, aregrounded. The value of resistor '80 is also taken as R0, but it isshunted to ground through resistance 83 of value Rd/Z.

In the interest of simplifying the calculations, it will be assumed herethat the current-voltage characteristics in Figs. 3 and 4 of the diodesgo through the origin. This is a permissible assumption since it wasestablished in connection with the multiplier of Fig. 5 that if thediodes are reasonably alike in their characteristics the effect of thebias circuit 62, 63 is to cause the effects of the bias b to' vanish.That is, weshall take b1=b2=b3= b4==0.

The performance of this structure can be found from Equations 11 through14. At junction 10, Equation 11 applies unchanged except that b1=0. Thisgives i =(e.,,+z+y)/4e R (l9) Atjunction H, Equation 12 applies witha:=1l=b:=0

Then

32: eM ieMRO (20) At junction I2, we use Equation 13 with y=ba==0 and atjunction 13, Equation 14 applies with x=b4=0. Therefore Substituting(19) through (22) into (10), we get for the output of amplifier $0,

output 5 of amplifier 80. The switch 91 is operated to position 98 andopens up the feedback resistance 6| amplifier 60. Theswitch 84 isoperated to make contact with terminal 93 which connects to the junction61 of amplifier 60. The above operation of switch 94 thus connects agenerator 96 whose output is w volts, through the resistance 95 of valueRa ohms into the junction 61. It will be observed that the neteffectofoperating the switches 90 and 94 is to replace the voltage 1 by thevoltage 1) and to add to the current I a component w/Rz. Assuming thatthe bias circuit 82, 63 has been properly adjusted, so that thequantities bi through In may effectively be considered to have zerovalues, the average current entering the junction 61 isnow I ='l +t +i+i +w/R 24 The values of the currents Z1, 22, 23" and u" can be foundfrom Equations 11 through 14 if in these equations the biases in b4 areset equal to zero and y is replaced by 17. Then i= m+ m' 0 "7 (e,,,a;17)/fie R (e,,,-a:+z7) /4e,,.R

7. (e,,,+a:-t) /4e,,,R5 Substituting these, values into Equation 24gives a it It the amplifier has an eifective input impedance Zt and avoltage amplification A, the output voltage will be i l If the productof'the amplification A and input impedance e su fic a t larg t eme a 1degree of accuracy q 7 I e R0 E 2R3 37 We are at liberty to take R3 atany convenient value. If we choose Rs=Ro/2 th en I i I w and, as

. 7 z this current may also be expressed byv thus, although the currentfrom the source 96 is actually supplied to the input 01' amplifier to,

due to the feedback action, a current represent-,

ing the factor to is actually supplied to the rectifiers 50A, 503.

Fig. 7 shows a circuitfor computing a voltage proportional to the squareof the magnitude of another voltage. It is apparent that this functioncould be accomplished by the circuits of Fig. 5 or 6 by having the twosignals :0 and :11 equal. However, the circuit of Fig. 7 requires lessapparatus.

In this figure, all components labeled with numbers used in Figs. 5 and6 are identical with the components used in Figs. 5 and 6. The only newcircuit element introduced is the battery I00, connected throughresistance l0! of value R0 to junction 61. The positive pole of thisbattery is grounded'and it applies Eo volts to the resistor NH.Consequently it delivers a current -Eo/Ro amperes to the junction 61.

A comparison of Figs. and 7 and an examination of Equations ll and 12show that the time average of currents flowing from diodes 50A and 583into junction 61 are Consequently, the output voltage 3 of amplifier is28,. R0 The circuit of Fig. 7 may be used to generate a voltage whosemagnitude is proportional to the square root of the magnitude of asignal voltage 11;. To do this, switches 90, 94 and 91 are operated topositions 9|, 93 and 98, respectively. The operation of swflsch removesthe signal m and replaces it by 0 which comes into terminal 9| over lead92 from amplifier Bil. The operation of switch 94 connects in thevoltage to and the switching on 9'! opens up the feedback resistance 6|of amplifier 60. Consequently, the total current entering the junctionis the contribution from w and that given by Equation 30 with 5replacing :0. Thus i 26MB) If the amplifier 60 has an efiective inputimpedance Z8 and a voltage amplification A, the output voltage is l lThe currents through the learners-Y1} addup to form a current e 17 26,.R2 R0 and, as

ze Rn w-, R3

this current is equivalent to thus, although the source 96 is connectedto the input circuit of amplifier 60, due to the feedback action, acurrent representing the factor w is actually supplied to the rectifiers50A, 503.

The output 12 will be proportional to the square root of w provided thatthe latter is represented by a negative voltage. This restriction isbrought about by the fact that the output current i1, i2 are alwayspositive. If the connections to the plates and cathodes of diodes 59Aand 50B are interchanged, a similar argument shows that w must now bepositive." If w is apt to be of either sign, the circuit of Fig. 5 maybe modified by the provision of switch 90A. The operation of this switchto position BIA will apply 12 to amplifier 2i through lead 92A. Then, ifw is a negative voltage, current flow is from diodes 50A and 50B. If wis positive, the current flow will be from diodes 5IA and SIB.

What is claimed is:

1. An electrical network producing avoltage representing a function oftwo factors comprising sources of two voltages respectively representingthe factors, a generator of a recurrent voltage wave having a triangularvariation of amplitude withtime, an amplifier having a low impedanceinput circuit and an output circuit, an integrating device connectedacross said input circuit, a plurality of two pole asymmetricallyconductive devices having one pole connected to said in tegratingdevice, a plurality of resistors connecting said'generator respectivelyto'the other poles of said asymmetrically conductive devices, and otherresistors respectivelyconnected from said sources to the other poles ofsaid'devices.

2. An electrical computing-network including an amplifier having a lowimpedance input circuit and an output circuit, a capacitor connectedacross the input circuit, first and second rectifiers poled in thepassing direction connected to said capacitor, third and fourthrectifiers poled in the blocking direction. connected to said capacitor,sources of two voltages representing positive and negative values of afactor connected through resistors respectively to said first and thirdand to said second and fourth rectifiers, other sources of two voltagesrepresenting positive and negative values of another "factor connectedthrough resistors respectively to said first and fourth and to saidsecond and third rectifiers, and a generator of a recurrent voltage-wavehaving a triangular variation ;of' amplitude with time respectivelyconnected through resistors to all said rectifiers, the maximumamplitude. of said wave being not less than the sum of the amplitudes ofsaid factor voltages, wherebythe outputvoltage of said amplifierrepresents the product of the factors.

3. The combination in claim 2 with switching means for connecting theoutput circuit of the amplifier in place of one of thesources of factorvoltages, and connecting this source through a 12 resistor to thecapacitor,. whereby theaoutput voltage ofthe amplifierrepresents thequotien of the factors.v

4. The. combination in. claim 2 in which the rectifiers are diodes,.with a source of. current connected to said capacitor; and adjusted-..to neutralize the currents due to curvature of the diodecharacteristic. 5. An electrical computing network including anamplifier having a low impedance input circuit and an output circuit, acapacitor connected across the input circuit, first and secondrectifiers poled in the passing direction'conn'ected to said capacitor,third and fourth rectifiers poled'in the blocking direction connected tosaid capacitor, a source of a first voltage representing a first factor,resistors connecting the source of said first voltage to the first andthird rectifiers, a source of a second voltage representing a secondfactor, resistors connecting the source of said second voltage to thefirst and fourth rectifiers, a generator of a recurrent voltagewave-having a triangular variation of amplitude with time, resistorsconnecting said generator to all said rectifiers, and connections fromthe sources of said first and second voltages to said output circuit,whereby the output of said amplifier represents the product of saidfactors.

6. Thecombination in claim 5 in which the rectifiers are diodes, with asource of current, a potentiometer having a winding connected to saidsource and a brush connected to said capacitor and adjusted toneutralize the currents due to curvature of the diode characteristics.

'7. An electrical computing network 'including an amplifier having a lowimpedance input circuit and an output circuit, a capacitor connectedacross said input circuit, firstv and second rectifiers poled in thepassing direction connected to said capacitor, sourcesof two voltagesrepresenting positive and negative values of a factor respectivelyconnected through resistors to said rectifiers, a generator of arecurrent voltage wave havinga triangular variation of amplitude withtime, resistors connecting said generators to both said rectifiers,andasource of current connected to'said capacitor 'and 'ad justed toneutralize the current-flowing into'sai'd capacitor due to said wave,whereby the output voltage of said amplifier represents the square ofsaid factor. V

8. The combination in claim? with'switching means for connecting theoutput circuit of-said amplifier in place of said source, and forconnecting said source through a resistor to the capacitor, whereby theoutput voltage of the amplifier represents the square'root of thefactor.

9. The combination in claim '7 'in' which the rectifiers are diodes,with 'a source ofc'urrent connected to the capacitor and adjusted 't'o'neutralize the currents due to curvature of the diode characteristic. V

10. An electrical network producing a quantity of electrical energyrepresenting a function of two known factors comprising, a groundedgenerator of a recurrent electrical wave haying-1a conductive device's;et -"capacitor connected from one pole of each of said devices toground, a utilization circuit connected across said capacitor,

a. plurality of resistors respectively connected, from saidgeneratortothe other poles-bf said. device's, grounded sources of quantitiesof'el'ec trical energy respectively representing the factors, and otherresistors connected from said sources to the other poles of saiddevices.

11. The combination in claim 10 in which two of the asymmetricallyconducting devices are connected to the generator in the passingdirection and two of these devices are connected to the generator in theblocking direction, the sources produce quantities of electrical energyrespectively representing positive and negative values of both of thefactors, and the sources are connected through the other resistors tosupply quantities of energy representing positive values of the factorsto one of the passing devices, to supply quantities of energyrepresenting negative values of the factors to the other passing device,to supply quantities of energy representing the positive value of onefactor and the negative value of the other factor to one of the blockingdevices and to supply quantities of energy representing the negativevalue of the one factor and the positive value of the other factor tothe other blocking device.

12. The combination in claim 10 in which two of the asymmetricallyconductive devices are connected to the generator in the passingdirection and two of these devices are connected to the generator in theblocking direction, the sources produce quantities of electrical energyrespectively representing positive values of the factors, the resistorsall have the same resistance, both sources are respectively connectedthrough resistors to one of the passing devices, a resistor having halfthe resistance of the other resistors is connected from the otherpassing device to ground, the

ground, and connections from the sources directly to the utilizationcircuit.

13. The combination in claim 10 in which two asymmetrically conductivedevices are connected to the generator in the passing direction and thesources supply quantities of electrical energy representing positive andnegative values of one factor, a third source of electrical energy, anda resistor connecting said third source to said capacitor to supply acurrent having an amplitude equal to one-half the maximum amplitude ofthe wave from the generator.

14. The combination in claim 1 in which two of the asymmetricallyconductive devices are connected to the generator in the passingdirection and two of these devices are connected to the generator in theblocking direction, means are connected to the sources to producevoltages of opposite polarities representing positive and negativevalues of the factors, and the other resistors are connected from thesemeans to the devices to supply voltages of one polarity to one of thepassing devices, voltages of the other polarity to the other passingdevice, and voltages of opposite polarities to the blocking devices,whereby the output voltage of the amplifier represents the products ofthe factors.

15. The combination in claim 14 with an added resistor and switchingmeans for disconnecting one of said sources and connecting this sourcethrough said added resistor directly to the integrating device, and forconnecting the output of the amplifier in place of this source, wherebythe output voltage of the amplifier represents the quotient of thefactors.

16. The combination in claim 14 with an added resistor, and switchingmeans for disconnecting both of said sources, for connecting the outputof the amplifier in place of both sources, and for connecting the sourceof one of the factor voltages through the added resistor to theintegrating device, whereby the output voltage of the amplifierrepresents the square root of this factor.

References Cited in the file of this patent UNITED STATES PATENTS NumberName Date 2,285,038 Loughlin June 2, 1942 2,419,852 Owen Apr. 29, 19472,486,068 Shishini et a1 Oct. 25, 1949 2,551,740 Hills May 8, 19512,568,927 Morrison Sept. 25, 1951 2,580,740 Dickinson Jan. 1, 19522,587,193 Miller Feb. 26, 1952

